<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">i1</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="scmfsa_1.html#K5" title="SCMFSA_1:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="card_3.html#K10" title="CARD_3:func.10">product"</a> <span class="p2">(<span class="default"><a href="amistd_2.html#K4" title="AMISTD_2:func.4">AddressParts</a> <span class="p3">(<span class="default"><a href="ami_1.html#K21" title="AMI_1:func.21">InsCode</a> <span class="p4">(<span class="default"><a href="scmfsa_2.html#K11" title="SCMFSA_2:func.11">goto</a> <font color="Maroon" title="c1">i1</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmfsa_1.html#K5" title="SCMFSA_1:func.5">NAT</a> </span><br/>

<a NAME="E1:54"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="card_3.html#K10" title="CARD_3:func.10">product"</a> <span class="p2">(<span class="default"><a href="amistd_2.html#K4" title="AMISTD_2:func.4">AddressParts</a> <span class="p3">(<span class="default"><a href="ami_1.html#K21" title="AMI_1:func.21">InsCode</a> <span class="p4">(<span class="default"><a href="scmfsa_2.html#K11" title="SCMFSA_2:func.11">goto</a> <font color="Maroon" title="c1">i1</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="scmfsa10.html#T36" target="_self" title="SCMFSA10:th.36">Th36</a>, <a class="ref" href="scmfsa_2.html#T47" title="SCMFSA_2:th.47">SCMFSA_2:47</a></span>;<br/>
<a NAME="E2:54"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A1</font></span>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="card_3.html#K10" title="CARD_3:func.10">product"</a> <span class="p2">(<span class="default"><a href="amistd_2.html#K4" title="AMISTD_2:func.4">AddressParts</a> <span class="p3">(<span class="default"><a href="ami_1.html#K21" title="AMI_1:func.21">InsCode</a> <span class="p4">(<span class="default"><a href="scmfsa_2.html#K11" title="SCMFSA_2:func.11">goto</a> <font color="Maroon" title="c1">i1</font></span>)</span></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<div><span class="kw">hereby </span> <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a>,<a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="scmfsa_1.html#K5" title="SCMFSA_1:func.5">NAT</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="card_3.html#K10" title="CARD_3:func.10">product"</a> <span class="p2">(<span class="default"><a href="amistd_2.html#K4" title="AMISTD_2:func.4">AddressParts</a> <span class="p3">(<span class="default"><a href="ami_1.html#K21" title="AMI_1:func.21">InsCode</a> <span class="p4">(<span class="default"><a href="scmfsa_2.html#K11" title="SCMFSA_2:func.11">goto</a> <font color="Maroon" title="c1">i1</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>
<div class="add"><span class="kw">let </span><font color="Maroon" title="c2">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="card_3.html#K10" title="CARD_3:func.10">product"</a> <span class="p2">(<span class="default"><a href="amistd_2.html#K4" title="AMISTD_2:func.4">AddressParts</a> <span class="p3">(<span class="default"><a href="ami_1.html#K21" title="AMI_1:func.21">InsCode</a> <span class="p4">(<span class="default"><a href="scmfsa_2.html#K11" title="SCMFSA_2:func.11">goto</a> <font color="Maroon" title="c1">i1</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 implies <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="scmfsa_1.html#K5" title="SCMFSA_1:func.5">NAT</a>  )</span><br/><span class="kw">assume </span><a NAME="E1:54_1"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="card_3.html#K10" title="CARD_3:func.10">product"</a> <span class="p2">(<span class="default"><a href="amistd_2.html#K4" title="AMISTD_2:func.4">AddressParts</a> <span class="p3">(<span class="default"><a href="ami_1.html#K21" title="AMI_1:func.21">InsCode</a> <span class="p4">(<span class="default"><a href="scmfsa_2.html#K11" title="SCMFSA_2:func.11">goto</a> <font color="Maroon" title="c1">i1</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="scmfsa_1.html#K5" title="SCMFSA_1:func.5">NAT</a> </span><br/><a NAME="E2:54_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K5" title="CARD_3:func.5">pi</a> <span class="p1">(<span class="default"><a href="amistd_2.html#K4" title="AMISTD_2:func.4">AddressParts</a> <span class="p2">(<span class="default"><a href="ami_1.html#K21" title="AMI_1:func.21">InsCode</a> <span class="p3">(<span class="default"><a href="scmfsa_2.html#K11" title="SCMFSA_2:func.11">goto</a> <font color="Maroon" title="c1">i1</font></span>)</span></span>)</span></span>)</span>,1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:54"><span class="lab"><font color="Green" title="E51">A1</font></span></a>, <a class="ref" href="card_3.html#T93" title="CARD_3:th.93">CARD_3:93</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c3">g</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><span class="kw"> such that </span><br/><a NAME="E4:54_1"/><span class="lab"><font color="Green" title="E52">A2</font></span>: 
<font color="Maroon" title="c3">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="amistd_2.html#K4" title="AMISTD_2:func.4">AddressParts</a> <span class="p1">(<span class="default"><a href="ami_1.html#K21" title="AMI_1:func.21">InsCode</a> <span class="p2">(<span class="default"><a href="scmfsa_2.html#K11" title="SCMFSA_2:func.11">goto</a> <font color="Maroon" title="c1">i1</font></span>)</span></span>)</span>
 <span class="kw">and </span><br/><a NAME="E5:54_1"/><span class="lab"><font color="Green" title="E53">A3</font></span>: 
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="card_3.html#D6" title="CARD_3:def.6">CARD_3:def 6</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c4">I</font> being   <a href="ami_1.html#NM2" title="AMI_1:NM.2">Instruction</a> of <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a> <span class="kw"> such that </span><br/><a NAME="E7:54_1"/><span class="lab"><font color="Green" title="E54">A4</font></span>: 
<font color="Maroon" title="c3">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="amistd_2.html#K3" title="AMISTD_2:func.3">AddressPart</a> <font color="Maroon" title="c4">I</font>
 <span class="kw">and </span><br/><a NAME="E8:54_1"/><span class="lab"><font color="Green" title="E55">A5</font></span>: 
 <a href="ami_1.html#K21" title="AMI_1:func.21">InsCode</a> <font color="Maroon" title="c4">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_1.html#K21" title="AMI_1:func.21">InsCode</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K11" title="SCMFSA_2:func.11">goto</a> <font color="Maroon" title="c1">i1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:54_1"><span class="lab"><font color="Green" title="E52">A2</font></span></a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c5">i2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="scmfsa_1.html#K5" title="SCMFSA_1:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E10:54_1"/><span class="lab"><font color="Green" title="E56">A6</font></span>: 
<font color="Maroon" title="c4">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmfsa_2.html#K11" title="SCMFSA_2:func.11">goto</a> <font color="Maroon" title="c5">i2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:54_1"><span class="lab"><font color="Green" title="E55">A5</font></span></a>, <a class="ref" href="scmfsa_2.html#T47" title="SCMFSA_2:th.47">SCMFSA_2:47</a>, <a class="ref" href="scmfsa_2.html#T59" title="SCMFSA_2:th.59">SCMFSA_2:59</a></span>;<br/><a NAME="E11:54_1"/>
<font color="Maroon" title="c3">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c5">i2</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:54_1"><span class="lab"><font color="Green" title="E54">A4</font></span></a>, <a class="txt" href="#E10:54_1"><span class="lab"><font color="Green" title="E56">A6</font></span></a>, <a class="ref" href="scmfsa10.html#T23" target="_self" title="SCMFSA10:th.23">Th23</a></span>;<br/><a NAME="E12:54_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">i2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:54_1"><span class="lab"><font color="Green" title="E53">A3</font></span></a>, <a class="ref" href="finseq_1.html#D8" title="FINSEQ_1:def.8">FINSEQ_1:def 8</a></span>;<br/><span class="kw">hence </span><a NAME="E13:54_1"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="scmfsa_1.html#K5" title="SCMFSA_1:func.5">NAT</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div>
<span class="kw">end;</span></div>

<span class="kw">let </span><font color="Maroon" title="c2">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="scmfsa_1.html#K5" title="SCMFSA_1:func.5">NAT</a>  or <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="card_3.html#K10" title="CARD_3:func.10">product"</a> <span class="p2">(<span class="default"><a href="amistd_2.html#K4" title="AMISTD_2:func.4">AddressParts</a> <span class="p3">(<span class="default"><a href="ami_1.html#K21" title="AMI_1:func.21">InsCode</a> <span class="p4">(<span class="default"><a href="scmfsa_2.html#K11" title="SCMFSA_2:func.11">goto</a> <font color="Maroon" title="c1">i1</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 )</span><br/>

<span class="kw">assume </span><a NAME="E4:54"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="scmfsa_1.html#K5" title="SCMFSA_1:func.5">NAT</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="card_3.html#K10" title="CARD_3:func.10">product"</a> <span class="p2">(<span class="default"><a href="amistd_2.html#K4" title="AMISTD_2:func.4">AddressParts</a> <span class="p3">(<span class="default"><a href="ami_1.html#K21" title="AMI_1:func.21">InsCode</a> <span class="p4">(<span class="default"><a href="scmfsa_2.html#K11" title="SCMFSA_2:func.11">goto</a> <font color="Maroon" title="c1">i1</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c3">x</font> = <font color="Maroon" title="c2">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="scmfsa_1.html#K5" title="SCMFSA_1:func.5">NAT</a>  ;<br/>
<a NAME="E6:54"/><span class="lab"><font color="Green" title="E52">A7</font></span>: 
<span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c3">x</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D8" title="FINSEQ_1:def.8">FINSEQ_1:def 8</a></span>;<br/>
<a NAME="E7:54"/>
 <a href="ami_1.html#K21" title="AMI_1:func.21">InsCode</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K11" title="SCMFSA_2:func.11">goto</a> <font color="Maroon" title="c1">i1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 6
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="scmfsa_2.html#T47" title="SCMFSA_2:th.47">SCMFSA_2:47</a></span>;<br/>
<a NAME="E8:54"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A8</font></span>: 
 <a href="ami_1.html#K21" title="AMI_1:func.21">InsCode</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K11" title="SCMFSA_2:func.11">goto</a> <font color="Maroon" title="c1">i1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_1.html#K21" title="AMI_1:func.21">InsCode</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K11" title="SCMFSA_2:func.11">goto</a> <font color="Maroon" title="c3">x</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="scmfsa_2.html#T47" title="SCMFSA_2:th.47">SCMFSA_2:47</a></span>;<br/>
<a NAME="E9:54"/>
 <a href="amistd_2.html#K3" title="AMISTD_2:func.3">AddressPart</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K11" title="SCMFSA_2:func.11">goto</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c3">x</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="scmfsa10.html#T23" target="_self" title="SCMFSA10:th.23">Th23</a></span>;<br/>
<a NAME="E10:54"/><span class="kw">then </span>
<span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c3">x</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="amistd_2.html#K4" title="AMISTD_2:func.4">AddressParts</a> <span class="p1">(<span class="default"><a href="ami_1.html#K21" title="AMI_1:func.21">InsCode</a> <span class="p2">(<span class="default"><a href="scmfsa_2.html#K11" title="SCMFSA_2:func.11">goto</a> <font color="Maroon" title="c1">i1</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:54"><span class="lab"><font color="Green" title="E53">A8</font></span></a></span>;<br/>
<a NAME="E11:54"/><span class="kw">then </span>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K5" title="CARD_3:func.5">pi</a> <span class="p1">(<span class="default"><a href="amistd_2.html#K4" title="AMISTD_2:func.4">AddressParts</a> <span class="p2">(<span class="default"><a href="ami_1.html#K21" title="AMI_1:func.21">InsCode</a> <span class="p3">(<span class="default"><a href="scmfsa_2.html#K11" title="SCMFSA_2:func.11">goto</a> <font color="Maroon" title="c1">i1</font></span>)</span></span>)</span></span>)</span>,1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:54"><span class="lab"><font color="Green" title="E52">A7</font></span></a>, <a class="ref" href="card_3.html#D6" title="CARD_3:def.6">CARD_3:def 6</a></span>;<br/>
<span class="kw">hence </span><a NAME="E12:54"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="card_3.html#K10" title="CARD_3:func.10">product"</a> <span class="p2">(<span class="default"><a href="amistd_2.html#K4" title="AMISTD_2:func.4">AddressParts</a> <span class="p3">(<span class="default"><a href="ami_1.html#K21" title="AMI_1:func.21">InsCode</a> <span class="p4">(<span class="default"><a href="scmfsa_2.html#K11" title="SCMFSA_2:func.11">goto</a> <font color="Maroon" title="c1">i1</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:54"><span class="lab"><font color="Green" title="E51">A1</font></span></a>, <a class="ref" href="card_3.html#T93" title="CARD_3:th.93">CARD_3:93</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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